Recall, that the formula utilized to calculate the future value of a lump sum is as follows:
Where:
FV = Future Value
PV = Present Value
r = rate
n = periods
Calculating the future value of a tax-deferred account incorporates the tax paid on the money when it is withdrawn during the final period. We can account for the taxes paid by adjusting the present value after it has been compounded by the specified rate and number of periods:
The addition of the quantity (1 – t) adjusts the future value in the final period by the tax that is owed.
Assume you have a present value of $1,000, the will grow at a rate of 7% for ten years, with an assumed tax rate of 30%. Plugging those values into the formula will yield the following:
Using Excel, we can model what happens during each of the ten periods:
| Year | PV | rate | FV | Tax (30%) |
| 1 | $ 1,000.00 | 7% | $ 1,070.00 | |
| 2 | $ 1,070.00 | 7% | $ 1,144.90 | |
| 3 | $ 1,144.90 | 7% | $ 1,225.04 | |
| 4 | $ 1,225.04 | 7% | $ 1,310.80 | |
| 5 | $ 1,310.80 | 7% | $ 1,402.55 | |
| 6 | $ 1,402.55 | 7% | $ 1,500.73 | |
| 7 | $ 1,500.73 | 7% | $ 1,605.78 | |
| 8 | $ 1,605.78 | 7% | $ 1,718.19 | |
| 9 | $ 1,718.19 | 7% | $ 1,838.46 | |
| 10 | $ 1,838.46 | 7% | $ 1,967.15 | |
| Tax | $ 590.15 | |||
| Net ATFV | $ 1,377.01 |
Notice how the tax is paid during the final period. In the United States, this is how the future value of a Traditional IRA would be calculated. We can represent the table above visually with the following chart:
Using an HP12C calculator, we can calculate the future value of a tax-deferred account with the following keystrokes:
[1000][PV]
[7][i]
[10][n][FV]
[.][7][*]
The formula can be rearranged as follows to find the present value of a tax-deferred account:
The present value of a tax-deferred account formula is usually only seen on tests which require you to calculate the present value of a tax-deferred account based on an initial investment an investor made in the past, given some current value in the future.
A copy of the Excel model can be found here.